Between me and my partner's alts, each avatar got a royal on first try. But it proved difficult for alts who didn't get it on first try, even after 3 or 4 pairs of cats.
I was beginning to think that the server was programmed to send you a royal only on your first birth in the stated period. We have about 40 cats between us, only with our very last box, we got a royal on an avatar who had tried numerous times.
I remember with the Dreams of Eire cat, we didn't even get a single one. Its not bad luck, its just randomness working against you